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To divide the polynomial [tex]\( P(x) = 6x^4 - 3x^3 + 29x^2 \)[/tex] by [tex]\( D(x) = 3x^2 + 13 \)[/tex] and express it in the form [tex]\( \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} \)[/tex], we use polynomial long division. Here’s a detailed, step-by-step solution.
1. Set up the division:
[tex]\[ \frac{6x^4 - 3x^3 + 29x^2}{3x^2 + 13} \][/tex]
2. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{6x^4}{3x^2} = 2x^2 \][/tex]
So the first term of [tex]\( Q(x) \)[/tex] is [tex]\( 2x^2 \)[/tex].
3. Multiply [tex]\( 2x^2 \)[/tex] by the denominator [tex]\( 3x^2 + 13 \)[/tex] and subtract this product from [tex]\( P(x) \)[/tex]:
[tex]\[ 2x^2 \cdot (3x^2 + 13) = (2x^2 \cdot 3x^2) + (2x^2 \cdot 13) = 6x^4 + 26x^2 \][/tex]
Subtract this from [tex]\( P(x) \)[/tex]:
[tex]\[ (6x^4 - 3x^3 + 29x^2) - (6x^4 + 26x^2) = -3x^3 + 3x^2 \][/tex]
4. Repeat the process with the new polynomial [tex]\(-3x^3 + 3x^2\)[/tex]:
Divide the leading term of the new polynomial by the leading term of the denominator:
[tex]\[ \frac{-3x^3}{3x^2} = -x \][/tex]
So, the next term of [tex]\( Q(x) \)[/tex] is [tex]\( -x \)[/tex].
5. Multiply [tex]\(-x\)[/tex] by the denominator [tex]\( 3x^2 + 13 \)[/tex] and subtract this product from [tex]\(-3x^3 + 3x^2\)[/tex]:
[tex]\[ -x \cdot (3x^2 + 13) = (-x \cdot 3x^2) + (-x \cdot 13) = -3x^3 - 13x \][/tex]
Subtract this from [tex]\(-3x^3 + 3x^2\)[/tex]:
[tex]\[ (-3x^3 + 3x^2) - (-3x^3 - 13x) = 3x^2 + 13x \][/tex]
6. Repeat the process with the new polynomial [tex]\(3x^2 + 13x\)[/tex]:
Divide the leading term of the new polynomial by the leading term of the denominator:
[tex]\[ \frac{3x^2}{3x^2} = 1 \][/tex]
So, the next term of [tex]\( Q(x) \)[/tex] is [tex]\( 1 \)[/tex].
7. Multiply [tex]\(1\)[/tex] by the denominator [tex]\( 3x^2 + 13 \)[/tex] and subtract this product from [tex]\( 3x^2 + 13x\)[/tex]:
[tex]\[ 1 \cdot (3x^2 + 13) = 3x^2 + 13 \][/tex]
Subtract this from [tex]\( 3x^2 + 13x\)[/tex]:
[tex]\[ (3x^2 + 13x) - (3x^2 + 13) = 13x - 13 \][/tex]
8. Since the degree of [tex]\( 13x - 13 \)[/tex] is less than the degree of [tex]\( D(x) \)[/tex], we stop here. This is our remainder [tex]\( R(x) \)[/tex].
Thus, the quotient [tex]\( Q(x) \)[/tex] is:
[tex]\[ Q(x) = 2x^2 - x - \frac{26}{3} \][/tex]
and the remainder [tex]\( R(x) \)[/tex] is:
[tex]\[ R(x) = 42x + \frac{338}{3} \][/tex]
So, the division of [tex]\( P(x) \)[/tex] by [tex]\( D(x) \)[/tex] in the form [tex]\( \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} \)[/tex] is:
[tex]\[ \frac{P(x)}{D(x)} = 2x^2 - x - \frac{26}{3} + \frac{42x + \frac{338}{3}}{3x^2 + 13} \][/tex]
1. Set up the division:
[tex]\[ \frac{6x^4 - 3x^3 + 29x^2}{3x^2 + 13} \][/tex]
2. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{6x^4}{3x^2} = 2x^2 \][/tex]
So the first term of [tex]\( Q(x) \)[/tex] is [tex]\( 2x^2 \)[/tex].
3. Multiply [tex]\( 2x^2 \)[/tex] by the denominator [tex]\( 3x^2 + 13 \)[/tex] and subtract this product from [tex]\( P(x) \)[/tex]:
[tex]\[ 2x^2 \cdot (3x^2 + 13) = (2x^2 \cdot 3x^2) + (2x^2 \cdot 13) = 6x^4 + 26x^2 \][/tex]
Subtract this from [tex]\( P(x) \)[/tex]:
[tex]\[ (6x^4 - 3x^3 + 29x^2) - (6x^4 + 26x^2) = -3x^3 + 3x^2 \][/tex]
4. Repeat the process with the new polynomial [tex]\(-3x^3 + 3x^2\)[/tex]:
Divide the leading term of the new polynomial by the leading term of the denominator:
[tex]\[ \frac{-3x^3}{3x^2} = -x \][/tex]
So, the next term of [tex]\( Q(x) \)[/tex] is [tex]\( -x \)[/tex].
5. Multiply [tex]\(-x\)[/tex] by the denominator [tex]\( 3x^2 + 13 \)[/tex] and subtract this product from [tex]\(-3x^3 + 3x^2\)[/tex]:
[tex]\[ -x \cdot (3x^2 + 13) = (-x \cdot 3x^2) + (-x \cdot 13) = -3x^3 - 13x \][/tex]
Subtract this from [tex]\(-3x^3 + 3x^2\)[/tex]:
[tex]\[ (-3x^3 + 3x^2) - (-3x^3 - 13x) = 3x^2 + 13x \][/tex]
6. Repeat the process with the new polynomial [tex]\(3x^2 + 13x\)[/tex]:
Divide the leading term of the new polynomial by the leading term of the denominator:
[tex]\[ \frac{3x^2}{3x^2} = 1 \][/tex]
So, the next term of [tex]\( Q(x) \)[/tex] is [tex]\( 1 \)[/tex].
7. Multiply [tex]\(1\)[/tex] by the denominator [tex]\( 3x^2 + 13 \)[/tex] and subtract this product from [tex]\( 3x^2 + 13x\)[/tex]:
[tex]\[ 1 \cdot (3x^2 + 13) = 3x^2 + 13 \][/tex]
Subtract this from [tex]\( 3x^2 + 13x\)[/tex]:
[tex]\[ (3x^2 + 13x) - (3x^2 + 13) = 13x - 13 \][/tex]
8. Since the degree of [tex]\( 13x - 13 \)[/tex] is less than the degree of [tex]\( D(x) \)[/tex], we stop here. This is our remainder [tex]\( R(x) \)[/tex].
Thus, the quotient [tex]\( Q(x) \)[/tex] is:
[tex]\[ Q(x) = 2x^2 - x - \frac{26}{3} \][/tex]
and the remainder [tex]\( R(x) \)[/tex] is:
[tex]\[ R(x) = 42x + \frac{338}{3} \][/tex]
So, the division of [tex]\( P(x) \)[/tex] by [tex]\( D(x) \)[/tex] in the form [tex]\( \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} \)[/tex] is:
[tex]\[ \frac{P(x)}{D(x)} = 2x^2 - x - \frac{26}{3} + \frac{42x + \frac{338}{3}}{3x^2 + 13} \][/tex]
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